Why doesn’t this logic work? (Where did I go wrong?) Essentially, I was wondering what percentage of numbers are prime. With only my logic and (very basic) mathematical understanding, I came up with this:
All prime numbers are odd, excluding 2.
There are the same amount of odd numbers as even numbers
So if I find what percentage of non-prime numbers are odd and subtract said percentage from 50, I should get the percentage of numbers that are prime.
Even * even = even.
Even * even = even.
Even * odd = odd.
Odd * even = odd.
Odd * odd = even.
Odd * odd = even.
Therefore 33.3% (recurring, of course) of non-prime numbers are odd.
Therefore 16.6% (recurring, of course) of numbers are prime.
I am aware that the actual percentage is far below this (possibly even zero?), but I’m not at all sure where I went wrong. Could somebody explain?
Thanks!
 A: Percentages are really meant for finite sets. But you did take a few wrong turns besides that.

All prime numbers are odd, excluding 2.

So far so good. No need to mention 0 and negative integers at this point.

There are the same amount of odd numbers as even numbers.

Yeah... well... now we need to be explicit that we're only looking at positive integers. Since each odd number can be matched up to just one immediately larger even number, we can indeed conclude that there are as many odd numbers as even numbers.

Even * even = even.

Correct.

Even * even = even.

Yeah, that's still correct, hasn't changed from the first time.

Even * odd = odd.

How do you get that? e.g., $2 \times 3 = 6$, and 6 is even, right?
But the biggest problem, as has already been pointed out in the comments, is that you can make the percentage whatever you want by making an arbitrary selection of numbers. For example:


*

*100% of primes are even if your selection is just {2}.

*50% of primes are even if your selection is {2, 3}.

*66% of numbers are prime if your selection is {1, 2, 3}.

*25% of numbers are prime if your selection is the first 100 positive integers.


And the selection doesn't have to consist of consecutive numbers.
Perhaps it would be more productive for you to review the following two facts: first, there are infinitely many prime numbers (Euclid proved this before Jesus was born); and second, the primes do thin out among the larger numbers; and third, almost all numbers are very large (the frivolous theorem of arithmetic).
